A monoid associated with a self-similar group action

We prove that there is a correspondence between self-similar group actions and the class of left cancellative right hereditary monoids satisfying the dedekind height property. The monoids in question turn out to be coextensive with the Zappa-Szép products of free monoids and groups, and the ideal structure of the monoid reflects properties of the group action. These monoids can also be viewed as ‘tensor monoids’ of covering bimodules, and also arise naturally from a double category associated with the action. There is also a correspondence between self-similar group actions and a class of inverse monoids, which are congruence-free when the actions are faithful; these inverse monoids arise naturally in the construction of the Cuntz-Pimsner algebras associated with the actions, and generalise the polycyclic monoids from which the Cuntz algebras are constructed. Finally, these results have the effect of correcting an error in a paper of Nivat and Perrot. 2000 AMS Subject Classification: 20M10, 20M50. 1 Self-similar group actions The results of this paper are closely related to a general theory originating in the pioneering papers of Rees [22] and Clifford [5], and subsequently developed by a number of authors [23, 17, 18, 16]. However the special case we consider is of independent interest, is developed from scratch, and has the added effect of shedding a new light on this general theory. The paper arose in the first